1. [latex]{\displaystyle\int }_{1}^{2}\frac{dx}{x}[/latex]; trapezoidal rule; [latex]n=5[/latex]

3. [latex]{\displaystyle\int }_{0}^{3}\sqrt{4+{x}^{3}}dx[/latex]; Simpson’s rule; [latex]n=3[/latex]

5. [latex]{\displaystyle\int }_{0}^{1}{\sin}^{2}\left(\pi x\right)dx[/latex]; midpoint rule; [latex]n=3[/latex]

11. Use the trapezoidal rule with four subdivisions to estimate [latex]{\displaystyle\int }_{2}^{4}{x}^{2}dx[/latex].

13. [latex]{\displaystyle\int }_{0}^{1}{\sin}^{2}\left(\pi x\right)dx[/latex]; trapezoidal rule; [latex]n=6[/latex]

15. [latex]{\displaystyle\int }_{0}^{3}\frac{1}{1+{x}^{3}}dx[/latex]; Simpson’s rule; [latex]n=3[/latex]

17. [latex]{\displaystyle\int }_{0}^{0.8}{e}^{\text{-}{x}^{2}}dx[/latex]; Simpson’s rule; [latex]n=4[/latex]

19. [latex]{\displaystyle\int }_{0}^{0.4}\sin\left({x}^{2}\right)dx[/latex]; Simpson’s rule; [latex]n=4[/latex]

21. [latex]{\displaystyle\int }_{0.1}^{0.5}\frac{\cos{x}}{x}dx[/latex]; Simpson’s rule; [latex]n=4[/latex]

25. Use the trapezoidal rule with four subdivisions to estimate [latex]{\displaystyle\int }_{0}^{0.8}{x}^{3}dx[/latex] to four decimal places.

27. Using Simpson’s rule with four subdivisions, find [latex]{\displaystyle\int }_{0}^{\frac{\pi}{2}}\cos\left(x\right)dx[/latex].

29. Given [latex]{\displaystyle\int }_{0}^{1}x{e}^{\text{-}x}dx=1-\frac{2}{e}[/latex], use the trapezoidal rule with 16 subdivisions to approximate the integral and find the absolute error.

31. Find an upper bound for the error in estimating [latex]{\displaystyle\int }_{4}^{5}\frac{1}{{\left(x - 1\right)}^{2}}dx[/latex] using the trapezoidal rule with seven subdivisions.

33. Find an upper bound for the error in estimating [latex]{\displaystyle\int }_{2}^{5}\frac{1}{x - 1}dx[/latex] using Simpson’s rule with [latex]n=10[/latex] steps.

35. Estimate the minimum number of subintervals needed to approximate the integral [latex]{\displaystyle\int }_{1}^{4}\left(5{x}^{2}+8\right)dx[/latex] with an error magnitude of less than 0.0001 using the trapezoidal rule.

37. Estimate the minimum number of subintervals needed to approximate the integral [latex]{\displaystyle\int }_{2}^{3}\left(2{x}^{3}+4x\right)dx[/latex] with an error of magnitude less than 0.0001 using the trapezoidal rule.

39. Use Simpson’s rule with four subdivisions to approximate the area under the probability density function [latex]y=\frac{1}{\sqrt{2\pi }}{e}^{\frac{\text{-}{x}^{2}}{2}}[/latex] from [latex]x=0[/latex] to [latex]x=0.4[/latex].

41. The length of one arch of the curve [latex]y=3\sin\left(2x\right)[/latex] is given by [latex]L={\displaystyle\int }_{0}^{\frac{\pi}{2}}\sqrt{1+36{\cos}^{2}\left(2x\right)}dx[/latex]. Estimate L using the trapezoidal rule with [latex]n=6[/latex].

45. The growth rate of a certain tree (in feet) is given by [latex]y=\frac{2}{t+1}+{e}^{\frac{\text{-}{t}^{2}}{2}}[/latex], where t is time in years. Estimate the growth of the tree through the end of the second year by using Simpson’s rule, using two subintervals. (Round the answer to the nearest hundredth.)

47. [T] Given [latex]{\displaystyle\int }_{1}^{5}\left(3{x}^{2}-2x\right)dx=100[/latex], approximate the value of this integral using the trapezoidal rule with 16 subdivisions and determine the absolute error.

49. The table represents the coordinates [latex]\left(x,\text{ }y\right)[/latex] that give the boundary of a lot. The units of measurement are meters. Use the trapezoidal rule to estimate the number of square meters of land that is in this lot.

51. The “Simpson” sum is based on the area under a ____.